MEN OF MATHEMATICS
line drawn through the fixed point and lying within the angle
formed by the two parallels. This apparently bizarre situation
is 'realized' by the geodesies on a pseudosphere.
For any everyday purpose (measurements of distances, etc.),
the differences between the geometries of Euclid and Lobat-
chewsky are too small to count, but this is not the point of
importance: each is self-consistent and each is adequate for
human experience. Lobatchewsky abolished the necessary
*truth' of Euclidean geometry. His geometry was but the first
of several constructed by his successors. Some of these substi-
tutes for Euclid's geometry - for instance the Riemannian
geometry of general relativity - are to-day at least as important
in the still living and growing parts of physical science as
Euclid's wass and is, in the comparatively static and classical
parts. For some purposes Euclid's geometry is best or at least
sufficient, for others it is inadequate and a non-Euclidean
geometry is demanded.
Euclid in some sense was believed for 2,200 years to have
discovered an absolute truth or a necessary mode of human
perception in his system of geometry. Lobatchewsky's creation
was a pragmatic demonstration of the error of this belief. The
boldness of his challenge and its successful outcome have
inspired mathematicians and scientists in general to challenge
other 'axioms' or accepted truths', for example the 'law* of
causalityr which, for centuries, have seemed as necessary to
straight thinking as Euclid's postulate appeared till Lobat-
chewsky discarded it,
The full impact of the LobatchewsMan method of challenging
axioms has probably yet to be felt. It is no exaggeration to call
Lobatchewsky the Copernicus of Geometry, for geometry is
only a part of the vaster domain which he renovated; it might
even be just to designate him as a Copernicus of all thought.